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Properties

Label 12-444e6-1.1-c0e6-0-1
Degree $12$
Conductor $7.661\times 10^{15}$
Sign $1$
Analytic cond. $0.000118369$
Root an. cond. $0.470728$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·7-s + 6·13-s − 27-s − 3·37-s + 3·49-s − 3·67-s − 3·79-s − 18·91-s + 3·103-s − 3·109-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + 179-s + 181-s + 3·189-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3·7-s + 6·13-s − 27-s − 3·37-s + 3·49-s − 3·67-s − 3·79-s − 18·91-s + 3·103-s − 3·109-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 21·169-s + 173-s + 179-s + 181-s + 3·189-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(0.000118369\)
Root analytic conductor: \(0.470728\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{6} \cdot 37^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3229096045\)
\(L(\frac12)\) \(\approx\) \(0.3229096045\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{3} + T^{6} \)
37 \( ( 1 + T + T^{2} )^{3} \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
13 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.43513087259533752621690969727, −6.05798112110947751284563553068, −5.85060427399910958311401958953, −5.73189329020752339186656700527, −5.69467496485834838357228796758, −5.61285821749543884238413263075, −5.58035420075195208393774082325, −5.02248410275126686901703221309, −4.73399647459845896232903938942, −4.50506601562562782448527920410, −4.38136277367718083949512563703, −4.16676088245441713148138730371, −3.86244551414909623103377178121, −3.72346803676097983848360538601, −3.53903613353404803245005779831, −3.43854977385681353137818426720, −3.30858617282751880688394370469, −3.11452213745379758835487578601, −3.10975008986571417225822215838, −2.63526788444988798891296065348, −2.28585030171626943257041257353, −1.73238038472723871136137090880, −1.69417690018706586725595872319, −1.28921091157841504566497699367, −1.15292438492964800010818308943, 1.15292438492964800010818308943, 1.28921091157841504566497699367, 1.69417690018706586725595872319, 1.73238038472723871136137090880, 2.28585030171626943257041257353, 2.63526788444988798891296065348, 3.10975008986571417225822215838, 3.11452213745379758835487578601, 3.30858617282751880688394370469, 3.43854977385681353137818426720, 3.53903613353404803245005779831, 3.72346803676097983848360538601, 3.86244551414909623103377178121, 4.16676088245441713148138730371, 4.38136277367718083949512563703, 4.50506601562562782448527920410, 4.73399647459845896232903938942, 5.02248410275126686901703221309, 5.58035420075195208393774082325, 5.61285821749543884238413263075, 5.69467496485834838357228796758, 5.73189329020752339186656700527, 5.85060427399910958311401958953, 6.05798112110947751284563553068, 6.43513087259533752621690969727

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.